jerry writes

There are other positions with the same symmetry characteristics as
the 4-spot. That is, there are other positions for which the
symmetry group contains sixteen elements. There are only three subgroups
of M containing sixteen elements, and the three subgroups are M conjugate.

these subgroups are the 2-sylow subgroups of M. one of sylow's
theorems states that any two p-sylow subgroups are conjugate.

one of these subgroups is the group of symmetries that preserve
the U-D axis. call this subgroup "P". (this is also the group
of symmetries of the intermediate subgroup of kociemba's algorithm.)

jerry asks about P-symmetric positions. coincidentally, i happened
to investigate these a few weeks back, and here's what i found:
(i calculated by hand, so i'd be grateful for any confirmation.)

there are 128 P-symmetric positions, 4 of which are M-symmetric.
they form a subgroup of the cube group (of course) which is
isomorphic to a direct product of 7 copies of C_2. in particular,
each such position has order 2 (or 1) as a group element. thus,
the answer to jerry's question

Call the 4-spot with all edges flipped t. Then, we certainly have
t'=t. Is this true of all positions whose symmetry group contains
sixteen elements?

is "yes". for what it's worth, this group of 128 positions can be
generated by the seven elements

superflip
pons asinorum
four spots
slice squared               ( U2 D2 )
eight flip                  ( FB UD RL FB UD RL )
four pluses                 ( R2 F2 R2 U'D F2 R2 F2 UD' )
four swapped corner pairs   ( D' B2 U'D F2 U2 L2 B2 L2 B2 U2 L2 F2 U )

however, these positions are not all locally maximal; for instance
U2 D2 is not.

mike